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Condensation of Cooper pairs and Cooper quartets in fermionic systems with multiple internal degrees of freedom

Aseem Talukdar

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

Condensation of Cooper pairs and Cooper quartets in fermionic systems with multiple internal degrees of freedom is studied.In this thesis work, I work on two major projects.On the first project I discuss Cooper pair condensation and while on the second I discuss Cooper quartet condensation.Due to the restrictions imposed by Pauli's principle, no two identical fermions can occupy a single quantum state.Therefore for electronic systems with two internal states, the maximum number of electrons that could be bound together is two.However; for a fermionic system having more than two internal states, it is possible that the bound state structure could be quite different.On my thesis I focus on systems that have four internal states.On one hand it is possible that the system will still undergo some types of pairing condensation, but there is also a possibility that the fermions will form a more complex structure where four fermions are bound together which we call a quartet.Physical systems where fermions can have four internal states include a system of spin-3 2 fermionic atoms and a two band electronic system.I look at possible two and four particle bound state structures in such systems.

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Condensation of Cooper pairs and Cooper quartets in fermionic systems with multiple internal degrees of freedom is studied.In this thesis work, I work on two major projects.On the first project I discuss Cooper pair condensation and while on the second I discuss Cooper quartet condensation.Due to the restrictions imposed by Pauli's principle, no two identical fermions can occupy a single quantum state.Therefore for electronic systems with two internal states, the maximum number of electrons that could be bound together is two.However; for a fermionic system having more than two internal states, it is possible that the bound state structure could be quite different.On my thesis I focus on systems that have four internal states.On one hand it is possible that the system will still undergo some types of pairing condensation, but there is also a possibility that the fermions will form a more complex structure where four fermions are bound together which we call a quartet.Physical systems where fermions can have four internal states include a system of spin-3 2 fermionic atoms and a two band electronic system.I look at possible two and four particle bound state structures in such systems.
The condensation of Cooper pairs, originating from the Fermi-surface instability due to a weakly attractive interaction between two fermions, opened a new frontier for exploring many-body physics in interdisciplinary contexts. In this work, we discuss the possible condensation of Cooper triples, which are three-body counterparts of Cooper pairs for three-component fermions with a three-body attraction. Although each composite trimer-like state obeys the Fermi-Dirac statistics, its aggregate can form a condensate at zero center-of-mass momentum in the presence of the internal degrees of freedom associated with the relative momenta of constituent particles of momenta close to the Fermi surface. Such condensation can be regarded as bosonization in infinite-component fermions. We propose a variational wave function for the condensate of Cooper triples in analogy with the Bardeen-Cooper-Schrieffer ground state, and we obtain the ground-state energy.
We theoretically discuss the possible condensation of Cooper triples, which correspond to a threebody version of Cooper pairs, in three-component Fermi systems with three-body attractive interactions.A macroscopic number of Cooper triples can occupy a zero center-of-mass momentum state in the presence of a Fermi surface of constituent particles, even though the three-body operator exhibits anti-commutation relation associated with the Fermi-Dirac statistics.Such a condensation with internal degrees of freedom is similar to bosonization in a system of infinite-component fermions.We propose a variational wave function for condensed Cooper triples and show that in the ground state, the condensed state is energetically favored compared to the normal state.Also, we discuss effects of the Fermi-surface distortion in a lattice system described by a three-component Hubbard model.
In this thesis we study the spin or pseudospin singlet pair condensation of two different kinds of polarized fermion systems.Using generalized BCS mean-field theories we study how pairing adapts to unequal spin or pseudospin populations.After briefly reviewing the basic physics of superconductivity in Chapter 2, in Chapter 3 the mean-field theory for electron-hole bilayer systems is derived to describe the condensation of excitons which is analogous to the Cooper pair condensation in superconductors.Self-consistent solution of the exciton system gap equation shows that the excitation energy spectrum is qualitatively the same as in superconductors.In Chapter 4 the role of the spin degree of freedom in the bilayer system is investigated by generalizing the two-component mean-field theory developed in Chapter vi 3 to four-component cases.The main consequence is that population polarization leads to ferromagnetism.The interplay between exciton condensation and spontaneous spin-order is the most important consequence of the presence of both spin and pseudospin degrees of freedom in excitonic condensates.In a sense that we explain in this Chapter, both normal and condensed fluids are present in the ferromagnetic excitonic state.Using the Rashba spin-orbit interaction model derived in the appendix, we show that an external electric field can alter the characteristics of the ferromagnetic condensate phase.The spin splitting by the spin-orbit interaction and its different spin state structures lead to qualitatively different magnetic properties for electron and hole layers.In Chapter 5 we turn our attention to a second class of polarized fermion systems that is of great current interest.A fully quantum mechanical treatment of a rotating fermion atom cloud is developed and implicit equations determining the critical temperatures for all center-of-mass Landau level pairings are obtained.In Chapter 6 the condition for the realization of higher center-of-mass Landau level pairing, which corresponds to FFLO state in spin split superconductors, is determined by calculating the critical temperatures for all possible pairing channels.It is shown that FFLO states can be realized in the strong interaction and low rotation frequency regimes in parameter space, where the pairing energy can survive the high polarization.
In the pair-correlation, or quasi-chemical equilibrium, approximation to statistical mechanics, there appears the possibility of a condensation phenomenon closely analogous to a Bose-Einstein condensation of correlated pairs of particles. However, the expansion used in earlier work to establish this phenomenon fails to yield an adequate approximation at and below the condensation point. In this paper, we give an alternative development, first formally for an arbitrary number of quantum states of the pair, then carry out in detail for the case of only one, or only two, quantum states of the pair. It is shown that, in spite of the Fermi-Dirac statistics of the particles making up the pairs, it is nevertheless possible to accomodate an arbitrarily large number of particles within a single pair state. Furthermore, if two quantum states of the pair are possible, and conditions are such that either the pair state would be occupied by a macroscopic number of particles if it alone were present, then the Pauli exclusion principle works in such a way that the lower one of the two pair states is occupied by a macroscopic number of particles, whereas the occupation number of the upper pair state remains independent of the volume. This result is not restricted to pairs whose mean separation exceeds their internal size, but applies also to pairs which completely overlap each other. The result is therefore of significance in the theory of superconductivity where the electron pairs have an internal size of the order of the Pippard coherence length, which is much larger than the mean separation of their centres of gravity.
Four-particle correlations in fermion systems at finite temperatures are investigated with special attention to the formation of a condensate. Instead of the instability of the normal state with respect to the onset of pairing described by the Gorkov equation, a new equation is obtained which describes the onset of ``quartetting.'' Applying to symmetric nuclear matter, below a critical density, the four-particle condensation ( $\ensuremath{\alpha}$-like quartetting) is found to be favored over deuteron condensation (triplet pairing). This pairing-quartetting competition is expected to be a general feature of interacting fermion systems, such as the exciton-biexciton system in excited semiconductors. Possible experimental consequences are pointed out.